Questions tagged [topology]
In mathematics, topology examines the properties of space (such as connectedness and compactness) that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.
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Torus shaped event horizon
Is there a solution to GR field equations for a rotating black hole that has a torus shaped event horizon? If so, when a craft flies through the torus, it can pass through a ring singularity while ...
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How does the Higgs field energy state relate to the tension of a Cosmic String?
I understand these cracks are massless objects, but if a string has an apparent mass from its warping of spacetime, how does this relate to the higher energy value of the Higgs field the Cosmic String ...
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Classical dynamics of point particles and fields on the torus: references
In simulations (e.g. molecular dynamics), a common trick to eliminate boundary effects is to use periodic boundary conditions. This forces us to understand the system's dynamics on a flat torus. In ...
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Prove the causal future of a compact subset of a globally hyperbolic spacetime is closed (Wald's Theorem 8.3.11)
In Wald's GR, theorem 8.3.11 states that
Let $(M,g_{ab})$ be a globally hyperbolic spacetime and let $K\subset M$ be compact. Then $J^+(K)$ is closed.
Let me present my own proof, first. I am not sure ...
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Fermions, Bosons, Anyons on a 2-manifold
My understanding of fermions, bosons, and anyons is that anyons are disallowed in 3+1 dimensions (or $n+1 | n\geq3$) because of the topology of spacetime. The paths of swapping two particles twice is ...
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Stabilization of "small" wormholes in an expanding space
Forget about wormhole-stabilizing fields and energies. Wheeler and Fuller's paper describe the expansion and subsequent collapse of a created wormhole.
Essentially they describe a created wormhole as ...
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Topological and non-topological defects?
The meaning of topological defect is only known intuitively to me. One explanation is it is some discontinuity in a system that cannot be removed.
But I would like to know the precise mathematical ...
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Properties of the interior of the domain of dependence $D(A)$
I have been reading the book Semi-Riemannian Geometry With Applications to Relativity by Barrett O'Neil, but I have some problems understanding the properties of $int(D(A))$, where $D(A)$ denotes the ...
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What is the topology of a de Sitter spacetime with multiple timelike dimensions?
(I believe that) de Sitter space is the only maximally symmetric Lorentzian spacetime, and that for $n$ spacetime dimensions, it has the hypercylindrical topology $\mathbb{R} \times S^{n-1}$.
This is ...
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Confusion on theorem 8.1.2 and corollary in Wald's GR book
In Wald's GR book theorem 8.1.2 says:
Let $(M,g_{ab})$ be an arbitray spacetime, and let $p \in M$. Then
there exists a convex normal neighborhood of $p$, i.e., an open set
$U$ such that for all $q,r ...
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How are topological entanglement and quantum entanglement related?
There are recurring articles about the relation between quantum entanglement and topological entanglement:
https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=topological+quantum+...
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What are $\mathcal{F}_g$ in string theory?
I was reading an article and came up on $\mathcal{F}_g$. Namely, it was in the following equation, $$\psi_{top} = \exp(\sum_g \mathcal{F}_g)$$ where I believe the $g$ denotes the genus of the topology ...
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Compactification in String Theory and Compactification in Topology are they the same thing?
In topology, there is a concept of compactification which is defined as follows.
A space $Z$ is a compactification of $X$ if $Z$ is compact Hausdorff and there exists an
embedding $j:X \rightarrow Z $ ...
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Assuming FLRW is correct, can it be 3-torus?
Assuming Friedmann–Lemaître–Robertson–Walker metric is correct, can space topology be 3-torus? I.e. are these two hypotheses compatible?
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Do branes in M-theory form closed structures or do they have an edge?
In M-theory, besides strings, branes play their role. Strings can be attached to them, for example when a number of branes in a stack rotate around a common axis and the strings between them get ...
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Hopfion with only in-plane vortex rather than skyrmion along the torus?
A simple torus-like hopfion with hopf charge $Q_H=1$ will typically exhibit a skyrmion at each slice cutting the toroidal circle. What if the skyrmion is replaced by an in-plane 2D vortex, i.e., we ...
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Topological charge change in QFT
Is it possible for the topological charge to change in quantum field theory?
The proofs in the following paper: Quantum soliton operators for vortices and superselection rules
are all based on the ...
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What's a Sewing Matrix?
I'm reading a paper on graphene that talks about these sewing matrices, but I don't understand their definition.
Upon researching it on the internet, I've found the term in other papers, so I assume ...
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Global space, and/or spacetime topological transitions in General relativity
One can consider the connected sum of two smooth n-manifolds $M_{1}$ and $M_{2}$ each with an embedded $\left(n-1\right)$ dimensional submanifold $V$. By deleting the interior of $V$ in both $M$s and ...
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Edge state protection in Chern insulator
I have a confusion about the nature of topologically protected boundary states in the Chern insulator. Since the Chern insulator does not require any symmetries to be present in the system, what is ...
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The reason why the Nielsen-Nimiya Theorem doesn't have to hold true in the Floquet system?
According to the Nielsen-Ninomiya (NN) theorem, under appropriate assumptions, the number of right-handed and left-handed particles must be equal in a lattice system. On the other hand, in recent ...
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How to quantify the idea that physical calculations of objects of close by geometry give same answer?
In many times in quick physics calculations, involving the geometry of a physical body, there is an assumption to simplfy the problem by considering the sample problem over a simpler geometry. ...
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Do spinor fields violate causality?
It is a theorem that spin structures on a spacetime $M$ exist iff the second Stiefel-Whitney class $w_2(M)=H^2(M, \mathbb{Z}_2)$ vanishes. I find this confusing for two reasons.
First, it implies that ...
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Abelian flat connection maps to zero
In an abelian gauge theory, a flat connection is an $A_\mu$ such that $F_{\mu\nu}=0$. I have to prove that these connections are equivalent to $0$, i.e. there is a gauge transformation that maps the ...
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Understanding the smoothness of Poincaré dodecahedral space
I've been trying to wrap my head around Poincaré Dodecahedral Space, aka the Poincaré homology sphere, which was suggested as a cosmological model after the release of WMAP cosmic microwave background ...
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Splitting of general pseudo-Riemannian Manifolds
Under what conditions splitting (e.g. $m + p$ foliation of a $m + p$ dimensional manifold) of a general pseudo-Riemannian manifold (with any arbitrary signature) possible? If it is too general then I ...
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How can I see the Calculation process of the skyrmion number?
I'm having a problem calculating a skyrmion number.
below equations are from Wikipedia - Magnetic skyrmion
I know that I should use the chain rule with the polar, spherical coordinate system.
But I ...
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Characterizing compactness of the Alexandrov topology in a spacetime
This is perhaps more of a soft question and on the mathematical side of things, but I'm struggling to find references and to formulate a precise argument. There's of course the chance that what I'm ...
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Is the Nambu-Goto action defined only for the torus?
For simplicity, I will use the Nambu-Goto action, but the following question would probably be the same for the Polyakov action.
According to David Tong's lecture notes on string theory, the Nambu-...
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What does Penrose mean when he talks about topology of spacetime?
Let us now set aside the question of the submicroscopic structure of space-time and concentrate, instead, on its large-scale properties. In this case, we may imagine that the smooth manifold picture ...
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Conditions for Bogoliubov-de Gennes Hamiltonian representation
The $H_{BdG}$ hamiltonian is described in topocondmat.org as follows:
here we can see that the submatrices along the diagonal are related as negative of complex conjugate of each other, I feel that ...
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Does physicists have a pre-conceived notion of continuity?
In many physics lecture on GR/ mathematical physics, one of the first things discussed is topology. I have seen many times that the reason for topology being discussed is that it's the weakest ...
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Will a light come back within finite years?
In this answer Javier said
Imagine the universe was the inside of a ball. We're 3D now, so no one is hiding any dimensions. This ball has a border, except it's not really a border. You should think ...
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Could the universe be a 4-ball?
I recently thought of the idea that the universe could be an infinite 4-ball. The Big Bang would be its centre, and time would be outward from its centre (one layer would be one point in time). I ...
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Can space only be infinite? [closed]
I have read before that if you could just go fast enough, as a thought experiment, and you move in a straight line, in any direction, that you eventually might reach the spot from which you started. I ...
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Reduction of the gravity gauge from various groups
As a gauge theory, the classic reduction for gravity is from the frame bundle to the Lorentz group,
\begin{equation}
GL(4) \to O(3,1)
\end{equation}
The associated configuration space of that ...
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What are the surfaces that contain an interior volume (space separating) called? Are they related to orientability?
I know that a "closed" surface is defined as a compact surface with no boundary. I don't have it clear if they have something to do with having an interior volume (completely enclosed volume)...
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Mathematics of the star-mesh transformation
I'm trying to understand the star-mesh transform from a mathematical perspective. This transformation removes a node and by definition, the topology of the network changes, but the resulting network ...
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What is a causal structure?
The notion of "causal structure" brings up many different notions in general relativity.
It is associated with the fiber bundle $\pi : \mathcal{C} \to M$ set of every light cone at every ...
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What is the correct domain of integration for the index of instantons? - $\mathbb{R}^4$ or $S^4$?
I posted the original question on Math SE but it seems like a more appropriate question for Physics SE:
https://math.stackexchange.com/q/4417225/
In calculating the instanton solutions for $SU(2)$ ...
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Photons as Observers and the Extended Real Number Line Topology
I am not a physicist. This is the first question I write in such a forum so if there are remarks on how I wrote it, I'll be happy to edit.
I am originally a mathematician with some interest in physics....
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Does Born-von-Karman boundary really change the topology?
As the title.
When studying the energy band of electrons by the famous A&M, I come to the confusion that the Born-von-Karman boundary seems to change the topology of materials. So whether the ...
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AdS$_4$ and $\mathbb{H}^4$: What is the difference between them?
This figure (source) shows the embedding of 4D hyperbolic space $\mathbb{H}^4$ and 4D de Sitter space dS$_4$ in 5D Minkowski space $\mathbb{M}^5$. $\mathbb{H}^4$ is a hyperboloid of two sheets and dS$...
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Emergent spacetime from a web of string world sheets?
Like many, I have deep conceptual difficulty understanding how an enormous amount of closed strings can become an effective classical spacetime that satisfies Einstein's equations. I appreciate that ...
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Quadruple index and the existence of corner state
I have been following this paper, which seems to discuss the connection between corner state and a quadruple index calculated $q(k_z)$ by equation (3) of it. At two special point $k_z=0 /\pi$, the ...
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Penrose diagrams for non-spherically symmetric spacetimes
As far as I have seen, Penrose diagrams are composed for spacetimes where there is spherical symmetry. The angular degrees of freedom are suppressed so as to understand the causal properties of ...
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How to calculate the variation of the metric on a compact manifold?
For example, given a torus with a modular parameter $\tau$ and an action
\begin{equation}
I=\frac{g}{2}\int_\mathcal{M} d^2 z \sqrt{-g}\ g_{ij}(z) \partial^i\phi \partial^j\phi
\end{equation}
...
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Does gap closing and reopening guarantee a non-trivial topological phase?
I know that a Dirac point carries a Chern number of $\pm\frac{1}{2}$ and when we have a gap closure at any point in our band structure we can transfer Chern numbers depending on at how many points we ...
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Topological classification of the Kane-Mele model?
Where should the Kane-Mele Model fall in the 10-fold way topological classification? I see that it is on a honeycomb lattice which is bipartite and thus has particle-hole symmetry. Going by that ...
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Quantum Field Theory on non-globally hyperbolic spacetimes?
In all the references I have found on QFT in curved spacetime, they treat only globally hyperbolic Lorentzian spacetimes, not Lorentzian spacetimes in general. Are there any references which discuss ...